#8069728

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#43095	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(10,14)(15,17)(16,22)(23,24)(25,26,31,33)(27,36,38,32)(28,34,37,30)(29,35,39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27,31,38)(26,32,33,36)(28,29,37,39)(30,35,34,40);(10,14)(15,17)(16,22)(23,24)(26,33)(30,34)(32,36)(35,40);(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,21)(22,23)(25,28)(26,34)(27,29)(30,33)(31,37)(32,40)(35,36)(38,39);(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,29)(26,35)(27,37)(28,38)(30,36)(31,39)(32,34)(33,40);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,30,31,34)(26,28,33,37)(27,35,38,40)(29,36,39,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976924);